State-Specified Protonium Formation in Low-Energy Antiproton–Hydrogen-Atom Collisions

We calculate state-specified protonium-formation cross sections in low-energy antiproton–hydrogen-atom collisions by solving the Chew-Goldberger–type integral equation directly instead of integrating the traditional differential scattering equation. Separating the incident wave from the total wave function, we calculate only the scattered outgoing wave propagated by the Green function. The scattering boundary condition is hence automatically satisfied without the tedious procedure of adjusting the wave function at the asymptotic region. The formed protonium atoms tend to be distributed in higher angular momentum $\ell$ and higher principle quantum number $n$ states as the collision energy increases. The present method has the advantage over the traditional ones in the sense that the required memory size and the computational time are much smaller, and accordingly the problem can be solved with higher accuracy.

Figure 1

Total protonium-formation cross sections for the incident energy below $n=2$ excitation threshold. The results (dashed curve) from Ref. 9 are also presented for comparison.

Figure 2

Total angular momentum dependent protonium-formation cross sections normalized to the total formation cross section at the incident energies of 2.72, 5.44, 8.16, and 10 eV, respectively.

Figure 3

$n$- and $\ell$-dependent protonium-formation cross sections at 10 eV incident energy.

Figure 4

(a) $\ell$- and (b) $n$-dependent protonium-formation cross sections at the incident energies 2.72, 5.44, 8.16, and 10 eV, respectively.